Problem 39 - Prealgebra

Question

Solve each equation by first finding the LCD for the fractions in the equation and then multiplying both sides of the equation by it.(Assume \(x\) is not 0 in Problems \(39-46\).) $$\frac{4}{x}=\frac{1}{5}$$

Step-by-Step Solution

Answer

The solution is \(x = 20\).

Step 1: Identify the Least Common Denominator (LCD)

To solve the equation \(\frac{4}{x} = \frac{1}{5}\), first identify the Least Common Denominator (LCD) of the fractions. The denominators are \(x\) and \(5\). Since \(x\) is variable, the LCD is simply \(5x\).

Step 2: Multiply Both Sides by the LCD

Multiply both sides of the equation by the LCD, \(5x\), to eliminate the fractions: \[ 5x \cdot \frac{4}{x} = 5x \cdot \frac{1}{5} \] This simplifies to: \[ 20 = x \]

Step 3: Verify the Solution

To verify, substitute \(x = 20\) back into the original equation: \[ \frac{4}{20} = \frac{1}{5} \] Simplify the left side: \[ \frac{1}{5} = \frac{1}{5} \] Both sides of the equation are equal, confirming that the solution \(x=20\) is correct.

Key Concepts

Understanding the Least Common Denominator

When it comes to equations involving fractions, one essential step is finding the "Least Common Denominator" (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder. This technique is valuable because it allows you to clear the fractions and work with whole numbers, which are easier to manage.
For example, in the equation \( \frac{4}{x} = \frac{1}{5} \), the denominators are \( x \) and 5. Here, \( x \) is a variable, so the LCD is just the product of these two terms: \( 5x \). By using the LCD, you essentially level the playing field, making it simpler to solve the equation.
Finding the LCD is particularly beneficial:

• It simplifies the equation by getting rid of fractions.
• It provides a consistent baseline for adjusting each term of the equation.

By mastering the concept of the least common denominator, you gain a crucial tool for solving equations efficiently.

Working with Fractions

Fractions can often seem daunting to work with, especially in equations. However, understanding their composition helps in simplifying these equations.
A fraction like \( \frac{4}{x} \) consists of two parts:

• The numerator (above the line), which is 4 in this case.
• The denominator (below the line), which is the variable \( x \).

The goal in solving equations with fractions is usually to eliminate them. This can be done by using the Least Common Denominator (LCD) to remove fractions entirely from the equation.
When fractions are involved, remember to:

• Identify the numerators and denominators first.
• Use the LCD to transform the equation into one without fractions.
• Be cautious with any variable in the denominator, as they imply restrictions on the values the variable can take (like \( x eq 0 \) if \( x \) is in the denominator).

Grasping the nuances of fractions makes handling complex equations significantly more straightforward.

Verifying Solutions

Once you have solved an equation, the final crucial step is to verify its solution. Verifying essentially checks whether the value you found actually satisfies the original equation.
For example, if you obtain \( x = 20 \) from \( \frac{4}{x} = \frac{1}{5} \), you substitute \( 20 \) back into the original equation to ensure it holds true:

• Substituting back gives \( \frac{4}{20} \).
• When simplified, \( \frac{4}{20} = \frac{1}{5} \), which matches the right side of the equation.

This verification is vital because:

• It confirms the correctness of your calculations and reasoning.
• It reveals any potential mistakes made during the process, allowing for corrections.

Verification builds confidence in the solution and assures that your results are accurate.